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Properties

Label 2-72-8.5-c17-0-77
Degree $2$
Conductor $72$
Sign $-0.999 + 0.0259i$
Analytic cond. $131.919$
Root an. cond. $11.4856$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (362. − 3.13i)2-s + (1.31e5 − 2.27e3i)4-s − 6.65e5i·5-s − 7.57e6·7-s + (4.74e7 − 1.23e6i)8-s + (−2.08e6 − 2.40e8i)10-s − 1.97e8i·11-s + 4.74e9i·13-s + (−2.74e9 + 2.37e7i)14-s + (1.71e10 − 5.95e8i)16-s − 3.37e10·17-s − 1.00e11i·19-s + (−1.51e9 − 8.71e10i)20-s + (−6.20e8 − 7.16e10i)22-s − 3.16e11·23-s + ⋯
L(s)  = 1  + (0.999 − 0.00866i)2-s + (0.999 − 0.0173i)4-s − 0.761i·5-s − 0.496·7-s + (0.999 − 0.0259i)8-s + (−0.00659 − 0.761i)10-s − 0.278i·11-s + 1.61i·13-s + (−0.496 + 0.00430i)14-s + (0.999 − 0.0346i)16-s − 1.17·17-s − 1.36i·19-s + (−0.0131 − 0.761i)20-s + (−0.00241 − 0.278i)22-s − 0.842·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0259i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.999 + 0.0259i$
Analytic conductor: \(131.919\)
Root analytic conductor: \(11.4856\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :17/2),\ -0.999 + 0.0259i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.9146843711\)
\(L(\frac12)\) \(\approx\) \(0.9146843711\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-362. + 3.13i)T \)
3 \( 1 \)
good5 \( 1 + 6.65e5iT - 7.62e11T^{2} \)
7 \( 1 + 7.57e6T + 2.32e14T^{2} \)
11 \( 1 + 1.97e8iT - 5.05e17T^{2} \)
13 \( 1 - 4.74e9iT - 8.65e18T^{2} \)
17 \( 1 + 3.37e10T + 8.27e20T^{2} \)
19 \( 1 + 1.00e11iT - 5.48e21T^{2} \)
23 \( 1 + 3.16e11T + 1.41e23T^{2} \)
29 \( 1 + 3.54e12iT - 7.25e24T^{2} \)
31 \( 1 - 2.76e11T + 2.25e25T^{2} \)
37 \( 1 + 2.12e13iT - 4.56e26T^{2} \)
41 \( 1 + 8.47e13T + 2.61e27T^{2} \)
43 \( 1 - 1.38e14iT - 5.87e27T^{2} \)
47 \( 1 + 8.21e13T + 2.66e28T^{2} \)
53 \( 1 + 3.16e14iT - 2.05e29T^{2} \)
59 \( 1 - 3.29e14iT - 1.27e30T^{2} \)
61 \( 1 - 6.60e14iT - 2.24e30T^{2} \)
67 \( 1 - 3.64e15iT - 1.10e31T^{2} \)
71 \( 1 + 9.19e15T + 2.96e31T^{2} \)
73 \( 1 + 5.44e15T + 4.74e31T^{2} \)
79 \( 1 - 1.06e16T + 1.81e32T^{2} \)
83 \( 1 + 9.38e15iT - 4.21e32T^{2} \)
89 \( 1 + 2.24e16T + 1.37e33T^{2} \)
97 \( 1 + 4.00e16T + 5.95e33T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.17970848125755684233330060885, −9.643522451555273932342866918202, −8.560322278138472960371462732714, −6.98604960862872583286582239640, −6.19998454905865196299823593695, −4.78322075947329244461206049055, −4.12350361371616832772913215880, −2.66301010641090979018966450738, −1.59218446270708812271936633493, −0.10836979728161982052809080381, 1.66244779580105028689828799473, 2.90484486253355533763018566957, 3.63313722311235821467940359858, 5.06832615876248005121954889067, 6.17400122341747240322878693333, 7.05000314637609487106979663069, 8.240546556788503865633869193877, 10.15559357201044273197141430984, 10.70547230475631231389114337208, 12.05279428525963830853608050357

Graph of the $Z$-function along the critical line